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Abstract

Background

The GABAergic system in the brain seems to be dysfunctional in various psychiatric
disorders. Many studies have suggested so far that, in schizophrenia patients, GABAergic
inhibition is selectively but consistently reduced in the prefrontal cortex (PFC).

Results

This study used a computational model of the PFC to investigate the dynamics of the
PFC circuit with and without chandelier cells and other GABAergic interneurons. The
inhibition by GABAergic interneurons other than chandelier cells effectively regulated
the PFC activity with rather low or modest levels of dopaminergic neurotransmission.
This activity of the PFC is associated with normal cognitive functions and has an
inverted-U shaped profile of dopaminergic modulation. In contrast, the chandelier
cell-type inhibition affected only the PFC circuit dynamics in hyperdopaminergic conditions.
Reduction of chandelier cell-type inhibition resulted in bistable dynamics of the
PFC circuit, in which the upper stable state is associated with a hyperactive mode.
When both types of inhibition were reduced, this hyperactive mode and the conventional
inverted-U mode merged.

Conclusion

The results of our simulation suggest that, in schizophrenia, a reduction of GABAergic
inhibition increases vulnerability to psychosis by (i) producing the hyperactive mode
of the PFC with hyperdopaminergic neurotransmission by dysfunctional chandelier cells
and (ii) increasing the probability of the transition to the hyperactive mode from
the conventional inverted-U mode by dysfunctional GABAergic interneurons.

Background

A number of studies have suggested alterations of the gamma-aminobutyric acid (GABA)
system in the brains of patients with schizophrenia (for reviews: [1-5]). The alteration of GABAergic neurotransmission in the cortex seems to be selective
for subpopulations of the interneurons [4-7]. Postmortem studies by Benes and colleagues reported decreased densities of interneurons
in layer II of the prefrontal cortex (PFC) and layers II-IV of the cingulate cortex
of patients with schizophrenia [8,9]. Possibly owing to its compensation, GABAA receptors were observed to be upregulated in layers II, III, V and VI in the PFC and
layers II and III in the cingulate cortex [10,11]. Decreased densities of the interneurons in the PFC and the cingulate cortex might
be restricted to the interneurons expressing calbindin; whether the densities of calretinin
(CR)- or parvalbumin (PV)-interneurons are reduced or not is still uncertain [6,12-14].

Analyses of postmortem brains of patients with schizophrenia have shown consistent
reduction of reelin, PV, and GAD67, the 67-kilodalton isoform of glutamic acid decarboxylase
[15-17]. Reelin is secreted preferentially by cortical GABAergic interneurons in layers I,
II and IV and binds to integrin receptors on dendritic spines of pyramidal neurons
or on GABAergic interneurons in layers III-V expressing the disabled-1 gene product
(DAB1) [15,18]. The expression of reelin mRNA was decreased in GABAergic interneurons in layers
I, II and IV of schizophrenia patients [19]. Because reelin plays a role in neuronal migration and synaptic plasticity in the
cerebral cortex [18,20,21], the reduction of reelin in schizophrenia would indicate a neurodevelopmental abnormality
that induces a GABAergic deficit in schizophrenia [1,21]. Reduced levels of mRNA for GAD67 in the dorsolateral prefrontal cortex (DLPFC) of
patients with schizophrenia suggest that GABA synthesis is reduced in schizophrenia
[22-25]. The reduction was detected in about 25-30% of the GABAergic interneurons in the
DLPFC [25,26]. Among many subtypes of GABAergic interneurons in the cortex, the PV-interneurons
contain basket cells and chandelier cells, which constitute 20-25% of GABAergic interneurons
in the primate DLPFC [28]. The GABAergic interneurons that show GAD67 mRNA reduction express PV [27], suggesting that the reduction is selective. Lewis and coworkers suggested that the
density of the GABA membrane transporter (GAT1)-immunoreactive axon cartridges of
chandelier cells was decreased by 40% in schizophrenic subjects compared to both normal
controls and subjects with other psychotic disorders [29,30]. They argued that the reduction was due to a decrease in the number of axon terminals
rather than the number of chandelier cells [29,30]. In contrast, the CR-positive GABAergic interneurons, which constitute about 50%,
were unaffected [3].

The regulation of GABAergic neurotransmission is critical for proper information processing
in the brain. For example, Goldman-Rakic and coworkers demonstrated that iontophoretic
application of bicuculline methiodide, a competitive antagonist of GABAA receptors, into the DLPFC of monkeys performing an oculomotor delayed response task
resulted in the destruction of spatial tuning of both pyramidal neurons and GABAergic
interneurons [31]. This has been reproduced in computational studies, which suggests that isodirectional
intracortical inhibition contributes to the stability of the cortical circuit and
cross-directional inhibition contributes to the spatial tuning or selectivity of working
memory to represent [32-34]. Therefore, the alteration of GABAergic neurotransmission in the cortex would cause
dysregulation of the circuit dynamics, resulting in the impairment of working memory
and other cognitive functions.

A recent neurophysiological study of rats [35] suggests that chandelier cells, whose spontaneous activity is fairly low, are reserved
to prevent excessive firing of neurons in the circuit. Chandelier cells are characterized
by their synapses on the axonal initial segment of pyramidal neurons and the reduction
of GABAergic neurotransmission in cortical circuits by this type of interneurons might
lead to disinhibitory overactivation of the cortex, such as epileptic activity [36]. Given that the density of the axon terminals of chandelier cells is reduced in schizophrenia,
as suggested by postmortem studies [30,37], one of the consequences of the circuit abnormality in schizophrenia would be hyperexcitability
of the cortex.

Early functional imaging studies reported reduced responses of the DLPFC or hypofrontality
in patients with schizophrenia [38-41]. Many recent studies suggest overactivation of the DLPFC during performing working
memory tasks [42-45] or both greater and less activation of subareas in the DLPFC [46,47]. The DLPFC would be basically hypodopaminergic, according to the dopamine (DA) hypothesis
of schizophrenia [48]. In this situation, the GABAergic inhibition in the DLPFC is not strong. Increasing
the DA release in the DLPFC increases glutamatergic neurotransmission through N-methyl-D-aspartate
(NMDA) receptors by D1 receptor stimulation. Then, the activity of the DLPFC increases
with the DA release in the DLPFC. Under hyperdopaminergic conditions, the GABAergic
inhibition becomes so strong that it highly suppresses noisy signal neurotransmission
in the DLPFC circuit [49]. The DLPFC activity thus shows an inverted-U shaped profile of the dopaminergic modulation
[50,51]. The profile would be sensitive to the strength of the GABAergic inhibition because
the decreasing phase of the inverted-U shaped curve critically depends on the GABAergic
inhibition in the DLPFC [49-51]. Therefore, if the GABAergic inhibition in the DLPFC is weakened, as has been observed
in schizophrenia, the activity of the DLPFC would be significantly different. In this
case, neurons in the DLPFC would exhibit hyperexcitation due to high NMDA currents
under hyperdopaminergic conditions [49,52].

Psychostimulants generally increase DA release from dopaminergic neurons [53]. Psychotic states induced by psychostimulants are accompanied by the focal activation
of the PFC, and the activity has a positive correlation with a psychotic symptom [54,55]. Therefore, hyperdopaminergic neurotransmission and hyperactivity would characterize
the PFC in acute psychotic states. The conventional inverted-U shape characteristic
of dopaminergic modulation of the PFC activity [50], however, does not predict this. It rather predicts hypoactivity of the PFC with
hyperdopaminergic neurotransmission. This unresolved issue would be an obstacle for
advancing our understanding of the circuit mechanisms of schizophrenia. Recently,
the circuit dynamics of the PFC under dopaminergic modulation has been studied using
a computational model of the PFC circuit [34,56-59]. This model predicts how the circuit dynamics of the PFC varies with D1 receptor
activation. The stability of the PFC circuit varies with the D1 receptor activation
when the operating point of the circuit moves along the inverted-U shaped curve. Using
this model, Tanaka and coworkers extended the range of the D1 receptor activation
to extremely high levels, and showed that hyperactivation of the PFC can occur under
hyperdopaminergic conditions (they termed this the 'H mode') [58]. Our study in this article uses essentially the same model and will explore the roles
of GABAergic inhibition in the regulation of such dynamics of the PFC circuit. The
result will show that 'chandelier cell-type inhibition' controls the H mode activity.
GABAergic interneurons other than chandelier cells do not regulate this hyperactive
mode effectively. Instead, these GABAergic interneurons regulate the conventional
inverted-U shape mode of PFC activity. With these results, we will discuss the roles
of GABAergic inhibition in the regulation and dysregulation of PFC circuit dynamics.
The aim of this article is to investigate how the GABAergic abnormalities observed
in the patients with schizophrenia alter the PFC circuit dynamics. Preliminary results
have been published in an abstract form [59].

Results

Mode diagram of the PFC

Equations (4) in Methods describe how the changes in the neuronal state variables
(xp, xc and xn for pyramidal neurons, chandelier cells and other GABAergic interneurons, respectively)
depend on the D1 receptor activation. To clarify the roles of the chandelier cells,
we first see the circuit dynamics of the PFC without chandelier cells. Figure 1 shows the dependence of the PFC activity on the D1 receptor activation. It is a contour
plot of dxp/dt, and the curves numbered 0 correspond to the equilibrium states of the pyramidal
neuron population. The equilibrium state is obtained mathematically from Equations
(4), by putting dxp/dt = dxc/dt = dxn/dt = 0, as

Figure 1.The mode diagram of the PFC with respect to dopaminergic modulation via D1 receptors. The vertical axis is the population activity of the pyramidal neurons in the PFC,
and the horizontal axis is the D1 receptor activation level (which is denoted by z in the text). Only the curves numbered 0 correspond to the equilibrium state of the
PFC circuit. See the text for the method of drawing of this diagram.

xp = τp [Wpp(z) fp(xp) - Wnp fn {τn(z)Wpn(z) fp(xp)}](1)

The above equation does not contain the term for the chandelier cells (or Wcp = 0) because we first see the circuit dynamics of the PFC without chandelier cells.
The relationship between xp and z in the above equation gives the mode diagram, which is identical to the curves for
the equilibrium states in Figure 1. The equilibrium state has two typical modes of the PFC activity in the different
range of D1 receptor activation, z. One is the inverted-U mode (1.0 <z < 4.3) and the other is the H mode (z > 6). There is a gap between these modes (4.3 <z < 6), in which the activity of the PFC is suppressed. Beyond z = 6, the PFC has two branches of activity. The upper branch is stable while the lower
branch is unstable, as shown below, meaning that the dynamics of the PFC circuit is
bistable. Therefore, once the PFC activity becomes higher than the unstable branch,
the activity increases to reach the upper stable branch, whereas PFC activity that
is lower than the unstable branch decreases to zero.

Analysis

One can see state transitions toward the stable equilibrium states by depicting nullclines
and fixed points at typical D1 receptor activation levels. Figure 2 shows the nullclines for the state variables of the pyramidal neurons and the GABAergic
interneurons other than chandelier cells, xp and xn. These nullclines are obtained by setting dxp/dt = dxn/dt = 0 as

Figure 2.Nullclines of the state variables of the pyramidal neurons and the GABAergic interneurons
other than chandelier cells for three different levels of D1 receptor activation. A: z = 3.0; B: z = 5.0; C: z = 7.0. The nullcline for the pyramidal neurons (dxp/dt = 0) is depicted in blue and the nullcline for the GABAergic interneurons (dxn/dt = 0) is depicted in green. The inset (A1) is an enlargement view of A. The circles
indicate stable fixed points and the crosses indicate unstable fixed points. The arrows
show the direction of state transition toward one of the stable fixed points.

(2)

These are the equilibrium conditions for the two populations of neurons. The intersections
of these nullclines indicate, therefore, the equilibrium states of the whole circuit
or the fixed points. The figure shows three different conditions, mentioned above;
i.e., the inverted-U mode (Figure 2A), the inactive state (Figure 2B), and the H mode (Figure 2C). The inverted-U mode has a single stable fixed point, indicated by a circle in Figures
2A and 2A1. The inactive state has no intersection between the two nullclines, so that only
the state xp = xn = 0 is stable. In the H mode condition, there are two intersections of the nullclines
or fixed points. Among these, the lower fixed point, indicated by a cross, is unstable,
whereas the higher fixed point, indicated by a circle, is stable. This stable fixed
point characterizes the H mode by hyperactivity of the PFC neurons.

Roles of GABAergic inhibition

We next investigate the roles of chandelier cells and other GABAergic interneurons.
We see how the changes in the strength of GABAergic inhibition alter the PFC activity.
The equilibrium condition of the PFC in this case is given by

The results are depicted in Figure 3, which are mode diagrams for different levels of GABAergic inhibition. Figure 3A is the same with Figure 1. In Figure 3B, the inhibition by the chandelier cells is increased, which moves the H mode away
from the inverted-U mode without altering the inverted-U mode profile. When the inhibition
by the GABAergic interneurons other than chandelier cells becomes weaker and the chandelier
cells are dysfunctional, the inverted-U mode and the H mode are connected (Figure
3C). Stronger inhibition, on the other hand, shrinks the inverted-U mode but does not
affect the H mode significantly (Figure 3D). This means that the inverted-U mode, but not the H mode, is sensitive to this type
of inhibition. A further increase in this type of inhibition eliminates the inverted-U
mode. In contrast, the H mode is robust against this type of inhibition; only the
chandelier cells can separate it from the inverted-U mode. Figure 4 shows the three-dimensional views of the temporal evolutions of these profiles. The
variations of the parameter values used in the simulation are summarized in Table
1.

Figure 4.Three-dimensional representations of DA modulatory landscapes with (B) and without
(A, C, and D) chandelier cells. The strength of the inhibition by other GABAergic interneurons are also varied (A:
1.0, B: 1.0, C: 0.95, and D: 1.06). Note that the onset of the H mode is very quick
(less than 100 ms), whereas the inverted-U mode profiles are very slow to evolve.
Even at t = 1000 ms, the profiles of the inverted-U mode have not reached the equilibrium states.
The profiles at equilibrium are shown in Figure 3.

Table 1. A summary of the variations of the parameter values of the two different types of
GABAergic inhibition used in the simulation.

Discussion

Chandelier cells vs other GABAergic interneurons

Our computational studies suggest that the dopaminergic modulation profile of PFC
activity is complex rather than just an inverted U. A remarkable thing is the possibility
of the existence of the H mode or hyperactive mode of the PFC with hyperdopaminergic
neurotransmission. Both this mode and the conventional inverted-U mode activity of
the PFC under the dopaminergic modulation would be regulated by GABAergic neurotransmission.
However, the simulation in this article suggests that these modes have different sensitivities
to different types of GABAergic inhibition. The H mode is sensitive to the GABAergic
inhibition by chandelier cells, whereas the inverted-U mode is sensitive to the inhibition
by GABAergic interneurons other than chandelier cells. The emergence of the H mode
is, therefore, critically dependent on the strength of the chandelier cell-type inhibition.
Stronger inhibition of this type puts the H mode away from the inverted-U mode. This
means that, when the GABAergic inhibition by chandelier cells is reduced, as suggested
in schizophrenia, the H mode is considered to be closer to the inverted-U mode than
in healthy controls. On the other hand, the profile of the inverted-U mode is critically
dependent on the inhibition by GABAergic interneurons other than chandelier cells.
If this type of inhibition is stronger, the inverted-U mode easily disappears. With
weaker inhibition of this type, in contrast, the profile of the inverted-U mode becomes
larger. If both types of inhibition are reduced, therefore, the inverted-U mode and
the H mode would merge into a single mode. As a result, the state of the PFC would
be able to move to the H mode from the inverted-U mode.

Transition to the H mode

The transition from the inverted-U mode to the H mode is illustrated by Figure 5. When the two modes are separated (Figure 5A), the inverted-U mode activity decreases as D1 receptors are activated further. Then,
it would be difficult to cross the gap to reach the H mode. Once they are connected
(Figure 5B), however, it would be much easier to reach the H mode from the inverted-U mode by,
for example, increasing the D1 receptor activation. The transition from the inverted-U
mode to the H mode would have important relevance to schizophrenia. First, the H mode
would be associated with psychotic states, as will be argued below. Second, chandelier
cells would prevent the occurrence of psychotic states by suppressing the H mode activity.
Third, weakening of the inhibition by other GABAergic interneurons increases the probability
of the transition to the H mode or vulnerability to psychosis. These would explain
the reason why schizophrenic brains are vulnerable to psychosis and are consistent
with the finding of the reduced GABAergic inhibition in the PFC of patients with schizophrenia.

Figure 5.Mode diagrams and state transition. A: When the chandelier cells are dysfunctional but the GABAergic inhibition by the
other interneurons is normal, the transition from the inverted-U mode to the H mode
hardly occurs because of a gap between the two modes. The gap becomes wider in the
existence of chandelier cells. B: When the chandelier cells are dysfunctional and
the GABAergic inhibition by the other interneurons is reduced, the two modes are connected,
so that the transition to the H mode would occur readily.

Psychosis

Schizophrenia

Functional magnetic resonance imaging (fMRI) studies of patients with schizophrenia
using a verbal fluency task showed that increasing task demand produced greater activation
of the PFC with higher error rates in psychotic states compared with remission [60]. A recent fMRI study suggested an association between reality distortion and hyperactivity
of the medial PFC of patients with schizophrenia or schizoaffective disorders [61]. Besides these, 'it is postulated that before experiencing psychosis, patients [with
schizophrenia] develop an exaggerated release of DA, independent of and out of synchrony
with the context' [62]. Downregulation of GABAergic neurotransmission in the PFC has consistently been associated
with schizophrenia [1-5,15]. These support our theory that psychotic states are induced by the transition to
the H mode due to reduced GABAergic inhibition in the PFC with hyperdopaminergic neurotransmission.

Substance-induced psychosis

Ketamine and amphetamines induced focal activation of the PFC in healthy subjects
[54,63,64] and in patients with schizophrenia [65]. In either schizophrenia or drug addiction, therefore, psychosis is associated with
selective or focal activation of the cortex [54,63-65]. NMDA antagonists, such as phencyclidine and ketamine, increase the extracellular
DA concentration in the PFC [66-68]. It has been suggested that acute administration of psychostimulants, such as amphetamines
and cocaine, increases the extracellular DA level significantly not only in the subcortical
areas but in the PFC [69,70]. A microdialysis study reported that intraperitoneal administration of 2 mg/kg of
amphetamine to rats induced six-fold increase in the baseline DA concentration in
the PFC [69], which could activate D1 receptors in the PFC. Recent studies reported that ketamine,
an NMDA antagonist, decreased the expression of PV and GAD67 in mice [71,72], suggesting reduced GABAergic inhibition in the PFC. Therefore, the underlying circuit
mechanism of substance-induced psychosis might be the same with schizophrenic psychosis;
that is, the transition to the H mode due to reduced GABAergic inhibition in the PFC
with hyperdopaminergic neurotransmission.

Dopamine-mediated mechanisms

Upregulation of D1 receptors

In contrast to acute administration, chronic administration of psychostimulants lowers
the extracellular concentration of DA in the PFC [73,74], which would induce sensitization of DA receptors. Similarly, the sensitization of
DA receptors would be expected in patients with schizophrenia. A positron emission
tomography (PET) study, using [11C]NNC 112 as a radiotracer, observed an increase
in the binding potential of D1 receptors in the PFC of schizophrenia patients [75]. This would reflect a chronically reduced extracellular DA concentration and an increase
in the density of D1 receptors. Upregulation or sensitization of D1 receptors might
be involved in schizophrenia. An increase in the DA releasability or the responsivity
of dopaminergic neurons has also been suggested [76,77]. These situations would increase the z value in the model, thereby increasing susceptibility to the H mode.

Stress

Acute stress increases DA turnover in the PFC, which leads to the impairment of cognitive
functions [78,79]. It seems that metabolic activity of dopaminergic neurons innervating the PFC is
increased selectively in the PFC [80]. The administration of the stressor FG 7142 also increases DA turnover in the PFC
[81,82]. Chronic stress induced hypodopaminergic states, and, again, impaired cognitive functions
[83]. In this case, Bmax or the density of D1 receptors in rat PFC was significantly increased (from 14.5 with
2.9 SD to 22.3 with 3.5 SD). Interestingly, either the hyperdopaminergic state or
the hypodopaminergic state with D1 upregulation could lead to the H mode, according
to the above arguments.

Epilepsy

People with epilepsy are susceptible to schizophrenia-like psychosis [84-86]. The association between epilepsy and schizophrenia-like psychosis has long attracted
much attention [87,88], and would be interesting to know the commonalities between epilepsy and schizophrenia
and the mechanisms underlying both diseases. Epilepsy is accompanied by excessive
excitation of neuronal circuits in the brain [89,90]. Many studies have suggested selective alterations in GABAA receptor subtypes in patients with epilepsy [91,92]. DeFelipe proposed the hypothesis that the chandelier cell is a key component of
cortical circuits in the establishment of epilepsy [36]. Links to dopaminergic mechanisms have also been suggested [93,94]. Using whole-cell recording and voltage-sensitive dye imaging techniques in the rat
PFC, Bandyopadhyay et al. [95] demonstrated that bath application of SKF 81297, a selective D1 receptor agonist,
enhanced spatiotemporal spread of activity in response to weak stimulation and previously
subthreshold stimulation resulted in epileptiform activity that spread across the
whole cortex. This result indicates that DA, via a D1 receptor-mediated mechanism,
enhances spatiotemporal spread of neuronal activity and lowers the threshold for epileptiform
activity in local circuits within the PFC. A rat study suggested that the supersensitivity
of the DA systems, which was developed in the chronic phase of the kainate-induced
temporal lobe epilepsy, was responsible for the genesis of epileptic psychosis [93]. The H mode hypothesis is consistent with all of these results.

Enhanced cortical inputs

Because of the bistable nature of the H mode, the occurrence of the H mode critically
depends on the strength of inputs. They are mediated by corticocortical or thalamocortical
afferents to the PFC, and would be modulated by several ways, including dopaminergic
modulation. It has also been suggested that DA has a sensorimotor gating function
in PFC and subcortical circuits [96-99]. In fact, many studies have reported deficits in the sensorimotor gating function
in patients with schizophrenia (for reviews: [98,100,101]) and, interestingly, also in amphetamine-sensitized animals [102]. When a dysregulated or unfiltered input is given to the PFC, the PFC would respond
to it with hyperactivity. Recent neurophysiological study in monkey reported an enhancement
of the response-period activity of DLPFC neurons, but no effect on delay-period activity,
by the stimulation of the D2 receptors in the DLPFC [103]. This may suggest that D2 receptors are involved in gating afferent input to the
DLPFC circuit for working memory and other cognitive functions. Moreover, if D2 receptors
are supersensitive [104,105], the H mode would more readily emerge because hyperactivation of D2 receptors could
contribute to the enhancement of the input to the PFC.

Conclusion

We have investigated how GABAergic inhibition by chandelier cells and other GABAergic
interneurons contribute to the regulation of neuronal activity in the PFC circuit.
The results show that the roles of the two different types of GABAergic inhibition
on PFC circuit dynamics are markedly different. The inhibition by GABAergic interneurons
other than chandelier cells effectively regulates the PFC activity with rather low
or modest levels of dopaminergic neurotransmission, which has an inverted-U shaped
profile of dopaminergic modulation and is associated with normal cognitive functions.
In contrast, the chandelier cell-type inhibition regulates the PFC activity with hyperdopaminergic
neurotransmission. Therefore, dysfunction of chandelier cells in the PFC would produce
the H mode, a "psychotic" hyperactive state with hyperdopaminergic neurotransmission.
Reduction of the inhibition by other GABAergic interneurons would make the transition
to the H mode more readily occur, thereby increasing vulnerability to psychosis.

Methods

Prefrontal Cortical Circuit Model

Our model of the PFC contains pyramidal neurons and GABAergic interneurons (Figure
6). The pyramidal neurons have recurrent connections or self-innervations. The two
populations of neurons are connected reciprocally. All of the neurons in the model
are assumed to be under dopaminergic modulation via D1 receptors; D1 receptor activation
changes the synaptic strengths from pyramidal neurons to both pyramidal neurons and
interneurons as well as the time constant for the interneurons [34,106]. The dopaminergic modulation via D1 receptors in this model is consistent with that
of Durstewitz et al. [107] but is a reduced one that is suitable for the present analysis with the firing rate
model.

Figure 6.A schematic diagram of the model. The PFC contains pyramidal neurons and GABAergic interneurons (chandelier cells
(C) and others (N)), which are connected reciprocally and have also self-innervations.
All populations of neurons are under dopaminergic modulation via D1 receptors. The
transient input to the pyramidal neurons triggers the dynamics of the circuit.

The pyramidal neurons receive a transient external input, which triggers the dynamics
of the circuit. Our model describes the activity of each population of neurons (either
pyramidal neurons or GABAergic interneurons) by a single state variable, which therefore
describes the population activity. The state equations for the population activities
are given by

(4)

where xp, xc and xn are the state variables for the pyramidal neuron population, the chandelier cell population,
and the population of the GABAergic interneurons other than chandelier cells, p, c and n denote the pyramidal neurons, the chandelier cells, and the GABAergic interneurons
other than chandelier cells, respectively, τp, τc and τn are the time constants of these neurons, Wij (i, j = p, c, n) is the synaptic efficacy from population i to j, and Icue is the transient external input to the pyramidal neuron population. The parameters
that depend on z are subject to dopaminergic modulation, where z is the D1 receptor activation (see below). The activation function is assumed to be
common to the populations of pyramidal neurons and GABAergic interneurons other than
chandelier cells:

GABAergic Interneurons

Spontaneous activity of chandelier cells is fairly low but they fire action potentials
at frequencies higher than other GABAergic interneurons when the overall cortical
excitation increases, suggesting that their role is to suppress excessive excitation
via their powerful inhibitory synapses on pyramidal neurons [35,108]. With their unique synapses on the axonal initial segment, chandelier cells would
increase their inhibitory effects when the GABA release from chandelier cell axon
terminals becomes coincident with spike generation of the postsynaptic pyramidal neurons.
This would require highly repetitive inputs from pyramidal neurons so that chandelier
cells can fire at a high rate. Therefore, the inhibitory effect by the chandelier
cell would increase sharply when the firing rate exceeds a certain threshold. We describe
this characteristic of inhibitory effect by chandelier cells simply with an activation
function

fc (x) = fmax tanh(x - x0)(6)

where x0 = 0.8 is the threshold above which the inhibition by the chandelier cell becomes effective.
Figure 7 shows the profiles of the activation functions for the populations of the chandelier
cells and other GABAergic interneurons. The difference in physiological properties
between these populations of neurons exists only in the thresholds in the activation
functions. For a network model consisting of different types of interneurons with
Hodgkin-Huxley models, refer to [109], which studied differential contributions to working memory representation in the
DLPFC. It would be interesting to see how the subtypes of interneurons affect the
profile of PFC activity under dopaminergic modulation.

Figure 7.The profiles of the activation functions of the chandelier cells (C) and the other
GABAergic interneurons (N) in the model.

Dopaminergic Modulation via D1 Receptors

The activation of D1 receptors affects the channel conductances, such as the conductances
of α-amino-3-hydroxy-5-methyl-4-isoxazole propionic acid (AMPA) and NMDA receptor-channels
(for reviews: [110,111]). These change the efficacy of glutamatergic signal neurotransmission (Wpp, Wpc and Wpn in this model). The excitability of the PFC inhibitory interneurons increases with
D1 receptor activation by decreasing the potassium-channel conductance [112]. This leads to a model in which the time constants of the interneurons, τc and τn, are assumed to decrease with D1 receptor activation [34,106]. Taken together, our model describes these effects by

(7)

where a, b and c are constants (a = 0.2, b = 0.4, c = 0.3).

Authors' contributions

ST carried out the design of the study, modeling, computer simulation, the analysis
of the results, and manuscript preparation.

Acknowledgements

This work was supported partly by the Sophia University Open Research Center grant.
The author acknowledges the discussions with Hiroaki Ebi toward the construction of
the computational model used in this study.